DRG ARRIVAL, N6843Q WAS CLEARED FOR AN NDB RWY 17 APCH. THE PLT'S LAST RADIO CALL WAS AT 1949 CST, WHEN HE ACKNOWLEGED RADAR SVC WAS TERMINATED & FREQ CHG WAS APPROVED. WHEN THE ACFT DID NOT ARRIVE, A SEARCH WAS INITIATED. AT 2335, AFTER FOG HAD LIFTED, THE ACFT WAS FND IN A PLOWED FLD APRX 1 MI NORTH OF THE RWY. AN INV REVEALED THE ACFT HAD CONTACTED THE GND IN A NOSE HI, WINGS LVL ATTITUDE WITH THE GEAR EXTENDED.INITIAL IMPACT OCCURRED (MAIN GEAR 1ST) WHILE ON A HDG OF 190 DEG, APRX 100 YDS RGT OF THE EXTENDED RWY CENTERLINE. THE NOSE GEAR THEN DUG INTO SOFT DIRT & THE ACFT FLIPPED OVER.THE AREA FORCAST HAD CALLED FOR OCNL CEILINGS BLO 1000' & VIS LESS THAN 3 MI. HOWEVER, SEARCH PSNL RPRTD THE SKY WAS OBSCD & THE VIS WAS ABOUT 1/16 MI WITH FOG. APRX 50 MI SW AT AMARILLO, TX, THE 1955 WX WAS IN PART: SKY ZERO OBSCD, VIS (RVR) 1600' WITH FOG. MINS FOR THE APCH WERE 600' CEILING & 1 MI VIS. THE PLT DID NOT REQUEST UPDATED WX INFO DRG THE FLT, NOR WAS HE GIVEN AVAILABLE UPDATED (AMA) WX INFO FOR THE AREA OF THE DESTN.